Active thrust

Active pressure calculation using Coulomb's method is based on global limit equilibrium theory of a system whose components are the wall and the wedge of homogeneous terrain behind the work assuming rough surface.

Where terrain is dry and homogeneous the pressure diagram is expressed linearly by the following:

Pt = KA · γt ·z

Thrust St is applied at 1/3 H with the value:

Representation of the failure wedge on the back of the wall

Values of Ka

δ < (β-ϕ-ε) according to Muller-Breslau

Where:

γt	Soil unit weight;
β	Inclination of the inner wall with respect to the horizontal plane passing through the footing;
φ	Soil angle of shearing resistance;
δ	Soil-wall friction angle;
ε	Inclination of the ground surface from the horizontal, positive if counterclockwise;
H	Wall height.

Active pressure calculation according to Rankine

If ε = δ = 0 and β = 90° (wall with smooth surface and backfill with horizontal surface) thrust St is simplified to:

that coincides with Rankine's equation for gives active pressure computation where backfill is horizontal.

Effectively Rankine used the same hypothesis as Coulomb except that he ignored wall-soil friction and cohesion. Rankine's expression for KA in general form is as follows:

Active pressure calculation according to Mononobe & Okabe

Mononobe & Okabe's evaluation of active thrust concerns thrust in seismic states with pseudo-static method. This is based on global limit equilibrium theory of a system whose components are the wall and the wedge of homogeneous terrain behind the work and involved in the failure in an artificial computation configuration in which ε, field level inclination angle with respect to the horizontal, and the angle β - inclination of internal wall surface to the horizontal -are increased by an amount θ such that:

where kh is the horizontal seismic coefficient and kv is the vertical seismic coefficient.

Where no specific studies exist, coefficients kh an kv should be calculated as:

where S·ag is the maximum seismic acceleration in the various categories of the stratigraphic profile.

Factor r can take the value r=2 where the work is one of some flexibility (eg. gravity walls). In all other cases (stiff reinforced concrete walls, reinforced concrete walls on piles or with anchors, fixed head walls - basement walls) it should be set to r=1.

Effect due to cohesion

Cohesion introduces negative constant pressures equal to:

As it is not possible to calculate a priori the thrust reduction induced in the thrust by cohesion a critical height Zc has been calculated as:

where

Q = Load acting on the backfill.

If Zc< 0 the effects may be applied directly as a decrease whose value is:

applied at H/2.

Uniform load on embankment

A load Q, uniformly distributed on the ground surface generates constant pressures as:

Integrating, a thrust Sq:

Applies at H/2, indicating with KA the active thrust coefficient according to Muller-Breslau.

Active thrust in seismic state

In seismic state the calculation force exercised by the embankment on the wall is given by:

where:

H = wall height

kv = vertical seismic coefficient

γ = unit weight

K = coefficients of total active thrust (static + dynamic)

Ews = hydrostatic thrust of water

Ewd = hydrodynamic thrust

For impermeable soils, hydrodynamic thrust Ewd = 0, but a correction on evaluation of the angle q in Mononobe & Okabe's formula is made as follows:

In highly permeable soils in seismic states, the same correction is applied but hydrodynamic thrust assumes the following value:

where H' is the height of the groundwater table (Gwt) from the base of the wall.

Hydrostatic thrust

Gwt whose surface is at height Hw from the base of the wall generates hydrostatic pressures normal to its surface that, at depth z, are expressed as:

Pw(z) = γw · z

With a resultant equal to:

Sw = 1/2 · γw· H²

The thrust of the submerged terrain can be obtained substituting γt with γ't (γ't = γsaturated - γw) , effective weight of submerged material.

Passive resistance

In homogeneous soil a linear diagram of pressures results:

integrating is obtained the passive thrust:

having:

(Muller-Breslau) with d limit values:

δ< β-φ-ε

The expression of Kp according to Rankine assumes the following form: